Variable Moment Flywheel

ABSTRACT

A method for storage of excess energy which would otherwise be lost, the regulation of angular velocity, and prevention of excessive velocities is disclosed. The device consists of a bowl shaped container, divided into sections by radially oriented vertical walls, which holds a fluid (any appropriate liquid or set of small solid particles), and spins on its vertically oriented axis at various angular velocities. The floor of the device is formed in successive shapes of bowls and shelves, which allows for a kind of “gearing”. The invention allows more and more energy to be input into the device while the angular velocity is regulated within a particular range. A typical embodiment of the invention would include its attachment by a shaft at the axis to a vertical axis wind turbine.

CROSS-REFERENCE TO RELATED APPLICATIONS

Provisional Application 62/133,440 Mar. 15, 2015; Non-ProvisionalApplication 15/062,930 Mar. 7, 2016 (parent patent application)

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable.

REFERENCE TO SEQUENCE LISTING, A TABLE, OR A COMPUTER PROGRAM LISTINGCOMPACT DISC APPENDIX

Not applicable.

TECHNICAL FIELD

The present invention relates generally to a method and a flywheelapparatus for absorbing and storing excess fluctuating energy and powerby varying the moment of inertia of the flywheel.

BACKGROUND

All devices designed to convert the kinetic energy of wind intomechanical and/or electrical energy, commonly called “wind turbines”,need to contend with the fluctuating nature of the wind. All existingwind turbine devices have to contend with the impact of varying angularvelocity on the voltage of the electricity produced. Variousaccommodations have been conceived to deal with the effects of thesevariations. These innovations include: variable speed electricalgenerating systems, blade pitch angle control, overspeed electronicbreaking protection, diversion load resistors, mechanical disk brakes,electronic angular velocity moderation keyed to wind speed, anemometersto measure wind speed, and various, innumerable other control devices.With all of these devices, the excess power of the wind is lost and isnot used to generate electrical or mechanical energy.

A fixed speed wind turbine generator system must be very mechanicallyrobust, adding to cost. Because the rotor speed cannot be varied,fluctuations in wind speed translate directly into drive train torquefluctuations, causing higher structural loads than with variable speedsystems.

A variable speed wind turbine generator system is necessarily morecomplicated than the fixed speed systems. The variable speed systemincorporates a doubly fed induction generator, a wound field synchronousgenerator, or a permanent magnet synchronous generator. These morecomplicated designs require electronic power converters which reduceelectrical efficiency and add to costs.

Additionally, “lift type” vertical axis wind turbines (“VAWT”) have tocontend with the problem of stall caused by rapid change of attack anglewhich results from excessive angular velocity.

The present invention is designed to convert the excess power of thewind into increased moment of inertia and potential energy, and therebyincreased storage of energy, which, when the wind ebbs, is convertedinto the kinetic energy of the rotor which then generates electricenergy. Additional benefits include regulation of angular velocity andprevention of excessive velocities. With the proper specific design, theinvention can control the angular velocity within the parametersrequired by the particular wind turbine and charging system.

The invention, with the proper specific design, can be used with anywind turbine of any size and any orientation. It can be used with ahorizontal axis wind turbine by incorporating a worm gear or somethingcomparable to allow for the vertical orientation of the presentinvention. It can also be used in other applications in which it isuseful to smooth the fluctuations of energy input, to store excessenergy, and to regulate the angular velocity of any mechanicalapparatus.

BRIEF SUMMARY OF THE INVENTION

The device consists of a bowl shaped container which holds a fluid. Thecontainer is horizontally oriented and spins on its vertically orientedaxis at various angular velocities. The bowl is divided into sections byradially oriented vertical walls. As the container turns, the walls pushthe fluid around and prevent it from sliding along the essentiallyfrictionless floor of the bowl.

The fluid can be any appropriate liquid or set of small spherical (ornon-spherical) solid particles.

The floor of the device is shaped in the radial direction in anincreasing angle to the horizontal approaching ninety degrees. This partof the device is called the first bowl. As the angle gets close toninety degrees, it rapidly drops to a much lower angle. This part of thedevice is called the first shelf. Then the angle increases, approachingninety degrees again. This part of the device is called the second bowl.This shape repeats as many times as is required by the specific turbine,charging system, and circumstances for which the invention is designed.This repeating shape of bowls and shelves allows for a kind of“gearing”: as excess energy is input into the device, it increases theangular velocity of the turbine, and thereby that of the invention. Themass is therefore pushed up the walls of the bowl in an outward radialdirection until it flows over the shelf and into the next bowl. Afterthe mass flows over the shelf, it rapidly flows outward, therebyincreasing the moment of inertia of the system. Conservation of angularmomentum then demands a reduction of angular velocity of the system. Anysubsequent increase in wind velocity can then increase the angularvelocity of the system again, thereby pushing the mass further outwardand up the walls of the next bowl. This cycle repeats as many times asthe system is designed to accommodate.

As the mass moves outward radially and upward, the angular momentum andstored energy of the system increases. When the wind ebbs, the angularvelocity decreases, thereby allowing the mass to fall toward the axis ofthe device, decreasing the moment of inertia and the energy of thesystem. The decrease in the moment of inertia and conservation ofangular momentum requires a subsequent increase in angular velocity. Thesystem therefore responds to gusts and lulls in the wind by respectivelyabsorbing and releasing energy to keep the angular velocity within aparticular range determined by the specific design of the turbine andcharging system.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph showing the height of the bowl of the invention as afunction of the radius of the bowl.

FIG. 2 is a vertical cross-section (8) of the device attached to a windturbine (1) by the central shaft (2) of the device. The drawing showsthe successive bowls and shelves of the device based upon the piece-wisefunction (FIG. 3) of a possible specific embodiment of the invention setforth under the Detailed Description of the Invention: 1st bowl (7), 1stshelf (6), 2nd bowl (5), 2nd shelf (4), and 3rd bowl (3). The windturbine and device is supported by a pole (9) which passes through thecentral shaft (2) of the device. The outer edge of each shelf (10) isjoined to the inner rim of each respective, succeeding bowl (10). Thehub (11) of the device is the base of the central shaft (2). The topedge of the device (12) is the outer rim of the uppermost bowl (3). Theempty interior space (13) is defined by the bowls (3, 5, 7) and shelves(4, 6).

FIG. 3 is the piece-wise function of a possible specific embodiment ofthe invention set forth under the Detailed Description of the Invention,and determines the shapes of the successive bowls and shelves of thedevice depicted in FIG. 2 in which the origin of the coordinate systemof the function is the horizontal center of bottom of the hub (11).

FIG. 4 is a top down view of the device showing the radially orientedvertical walls (14) dividing the empty interior space (13) in relationto the central shaft (2), the supporting pole (9), and the top edge ofthe device (12).

DETAILED DESCRIPTION OF THE INVENTION

Mathematics. The mathematics of the invention is derived as follows:Picture a bowl-shaped device with inclined sides spinning about itsaxis. In the device are radially oriented vertical walls. Also in thedevice is a fluid substance (mass) which moves up and down the sides.The fluid mass can be any appropriate liquid or set of small spherical(or non-spherical) solid particles. As the device spins on its axis,this mass is pushed around by the radially oriented vertical walls. Asthe angular velocity increases, the mass moves further out radially andup the inclined sides of the device. The total energy of the device isthe sum of the potential energy and kinetic energy.

The potential energy, U=mgh, where m is the mass, g is the accelerationdue to gravity, and h is the height of the mass from the base of thedevice.

The kinetic energy,

${K = {{\frac{1}{2}\omega^{2}r^{2}m} = {\frac{1}{2}\omega^{2}I}}},$

where m is the mass; ω is the angular velocity of the device and, byvirtue of the vertical walls in the device, the angular velocity of themass; r is the distance of the mass from the axis of the bowl; and I isthe moment of inertia of the system.

The moment of inertia, I=r²m. The invention is called a “variable momentflywheel” because the moment of inertia varies with the radius, thedistance of the mass from the axis of the bowl.

Moment Of Inertia Dilemma. As ω increases or decreases, so do r and h,and so do the kinetic and potential energies of the device. But, as ωincreases and r increases, so does the moment of inertia I, which, ifthe energy is constant, leads to a decrease of the angular velocity ω,thereby leading to a decrease in the radius r of the mass, which leadsto a decrease in the moment of inertia I, which decrease would lead toan increase in ω and r, ad infinitum. This dynamic would lead to a sortof yo-yo effect in which the radius of the mass, the angular velocity,the moment of inertia, and the kinetic and potential energies wouldincrease and decrease constantly, possibly increasing in frequency andviolence.

The design of a successful device requires a resolution of the abovedescribed dilemma. This dilemma is resolved by constraining thevariables ω, r and h (as derived from potential energy) into particularrelationships with each other. The specifics of these relationships willbe determined by the specific application of the invention. The generalforms of the equations are:

Radius as a function of angular velocity: r(ω)=k₁ω^(e) ¹ +C₁

Potential energy as a function of kinetic energy: U(K)=k₂K^(e) ² +C₂

Height as a function of radius (C₂=0 to simplify the mathematics):

h(r)=k ₃(r(ω))^(e) ³ +C ₃

The typical values for e₁, e₂, and e₃ are 1, 1 and 4, respectively. Inthe typical case,

${k_{3} = \frac{k_{2}}{2{gk}_{1}^{2}}},$

and, therefore, h(r)=k₃(r⁴−2r³C₁+r²C₁ ²)+C₃. (typical curve)

The values for the constants are determined by the specifics of theapplication. It is also possible to frame r as some other kind offunction of ω, and U as some other kind of function of K. The functionh(r) would then be derived from those two functions. However, themathematics are often more complicated and the results are not any moreuseful than the above equations.

Flatness. The importance of defining h(r) is that this functiondescribes the shape of the bowl of the device. One importantcharacteristic of the shape of the bowl, or the curve of the function,is its “flatness”. The typical curve is concave up. The flatter thecurve, the more unstable will be the position of the mass on the surfaceof the bowl. (A straight line or concave down curve will be whollyunstable and ineffective.) The limit to how flat the curve can be isdetermined when a mass can be placed anywhere on the bowl at a certainangular velocity, and the mass will stay put at that angular velocity. Alittle faster, and the mass will fly up the side. A little slower, andthe mass will fall to the bottom. This curve of maximum “flatness” isdefined by the following function:

${h(r)} = {{r\mspace{11mu} {\tan^{- 1}({kr})}} - {\frac{1}{2}{\ln \left( {1 + {k^{2}r^{2}}} \right)}} + C}$

(curve of maximum flatness).

This function is derived as follows: For an incline, gravitational forceis equal to the vertical component of the normal force: F_(g)=mg=N cosθ, where m is the mass, g is the acceleration due to gravity, N is thenormal force, and θ is the angle of the incline with the horizontal.Rearranging this equation, we obtain an expression for the normal force:

$N = {\frac{mg}{\cos \mspace{11mu} \varphi}.}$

Centripetal force is: F_(C)=mrω² , where r is the distance of the massfrom the axis of the device and ω is the angular velocity of the bowland the mass. This centripetal force is equal to the horizontalcomponent of the normal force, giving us: F_(C)=mrω²=N sin θ.

Substituting

$\frac{mg}{\cos \mspace{11mu} \varphi}$

for the normal force in this equation, we get:

${{mr}\; \omega^{2}} = {\frac{mg}{\cos \mspace{11mu} \varphi}\sin \mspace{11mu} {\varphi.}}$

Simplifying this equation and solving for θ yields:

$\varphi = {{\tan^{- 1}\left( \frac{r\; \omega^{2}}{g} \right)}.}$

A mass placed on an incline of angle θ at a particular radius r andangular velocity ω according to this equation represents a situation inwhich the mass will not move up or down the incline.

Since θ represents the slope of the bowl, the integral of this functionwill represent the shape of the bowl. If we keep co constant

$\left( {{{say}\mspace{14mu} k} = \frac{\omega^{2}}{g}} \right),$

θ is a function only of r.

Then θ(r)=tan⁻¹(kr) and the shape of the bowl for a particular angularvelocity is described by the following integral:

${\int{\varphi (r)}} = {{h(r)} = {{r\mspace{11mu} {\tan^{- 1}({kr})}} - {\frac{1}{2}{\ln \left( {1 + {k^{2}r^{2}}} \right)}} + C}}$

(curve of maximum flatness).

Since in the above equation

$k = \frac{\omega^{2}}{g}$

where ω is the angular velocity and g is acceleration due to gravity,and wind turbines may be designed to operate at a wide range of angularvelocities, the value for k may also vary widely. For example, one windturbine may be designed to operate at an angular velocity of 0.1 radiansper second, while another may be designed to operate at 30 radians persecond or even higher. The values of k at these two different angularvelocities would be

$k = {\frac{\left( {0.1\frac{rad}{s}} \right)^{2}}{9.81\frac{m}{s^{2}}} = {{.001}\frac{{rad}^{2}}{m}\mspace{14mu} {and}}}$${k = {\frac{\left( {30\frac{rad}{s}} \right)^{2}}{9.81\frac{m}{s^{2}}} = {91.7\frac{{rad}^{2}}{m}}}},$

respectively. Therefore, value of k used to determine the curve ofmaximum flatness may be any positive number, depending on the specificwind turbine system at issue.

The equation of the curve of maximum flatness,

${{h(r)} = {{r\mspace{11mu} {\tan^{- 1}({kr})}} - {\frac{1}{2}{\ln \left( {1 + {k^{2}r^{2}}} \right)}} + C}},$

indicates that when the radius is zero, r=0, the height will equal C,h(r)=C . This constant C indicates the height of the floor of the bowlabove (or below if C is negative) a fixed point at the axis of rotationof the device. Although it is easiest to locate that fixed point at thefloor of the bowl at the axis of rotation thereby setting C equal tozero, the fixed point may be chosen arbitrarily by the designer of aparticular system, and so the value of C could be any arbitrary number.

So, a mass placed at any point on a spinning bowl of this shape will notmove up or down as long as co remains at a specific constant value. Ifco increases, the mass will move rapidly to the outer edge of the bowl.If ω decreases, the mass will fall rapidly to the center of the bowl. Toallow for smoother operation of the invention, the shape of the bowl(the curve of the function) needs to be less “flat”, allowing for somestability of the mass on the side of the bowl. This is accomplished bythe typical equation and variations of it.

An example of a comparison between these two curves follows: Theequation of the typical curve in FIG. 1 is h(r)=0.433231·r⁴(k₃=0.433231,C₁=C₂=C₃=0), and the equation of the curve of maximum flatness in FIG. 1is

$\mspace{14mu} {{h(r)} = {{r\mspace{11mu} {\tan^{- 1}(r)}} - {\frac{1}{2}{{\ln \left( {1 + r^{2}} \right)}.}}}}$

(k=1, C=0).

Bowls and Shelves and Gearing. The floor of the device is shaped in theradial direction in an increasing angle to the horizontal approachingninety degrees. This part of the device of increasing angle is calledthe first bowl. As the angle gets close to ninety degrees, it rapidlydrops to a much lower angle. This lower angle part of the device iscalled the first shelf. Then the angle increases, approaching ninetydegrees again. This part of the device of increasing angle is called thesecond bowl. This shape repeats as many times as is required by thespecific turbine, charging system, and circumstances for which theinvention is designed.

This repeating shape of bowls and shelves allows for a kind of“gearing”: as excess energy increases the angular velocity of theturbine, and thereby that of the device itself, the mass is pushed upthe walls of the first bowl in an outward radial direction until itflows onto the first shelf. After the mass flows onto the first shelf,it rapidly flows outward, thereby increasing the moment of inertia ofthe system. Conservation of angular momentum then demands a reduction ofangular velocity of the system. Any subsequent increase in wind energycan then increase the angular velocity again, thereby pushing the massfurther outward and up the walls of the second bowl. This cycle repeatsas many times as the system is designed to accommodate. Each bowl isdesigned to operate within certain limits of angular velocity, but eachsuccessive bowl operates at higher levels of kinetic energy than theprevious bowl due to the increase in the moment of inertia of thedevice.

As the mass moves outward radially and upward, the angular momentum andstored energy of the system increases. When the wind ebbs, the angularvelocity decreases, thereby allowing the mass to fall toward the centerof the device, decreasing the moment of inertia and the energy of thesystem. The decrease of the moment of inertia and conservation ofangular momentum requires a subsequent increase in angular velocity. Thesystem therefore responds to gusts and lulls in the wind by respectivelyabsorbing and releasing energy to keep the angular velocity within aparticular range determined by the specific design of the turbine andcharging system.

The importance of defining h(r) is that this function describes theshapes of the bowls and shelves of the device. These shapes determinethe behavior of the invention. The origin of the coordinate system ofthe function is at the intersection of the axis and the base of thedevice. At the top edge of the first bowl, another function takes overto create the first shelf of the device. A third function, similar tothe original function h(r), takes over at the outer edge of the firstshelf to define the shape of the second bowl. This pattern repeats foras many bowls and shelves the system requires, resulting in a piece-wisefunction. This piece-wise function will typically take the followingform:

${h(r)} = \left\lbrack \begin{matrix}{{a \leq r \leq b},} & {{k_{3}r^{4}} + C_{3}} & {{first}\mspace{14mu} {bowl}} \\{{b < r \leq c},} & {{k_{4}\sqrt{r}} + {h(b)}} & {{first}\mspace{14mu} {shelf}} \\{{c < r \leq d},} & {{k_{5}r^{4}} + {h(c)}} & {2{nd}\mspace{14mu} {bowl}} \\{d\; \ldots} & {{k_{6}\sqrt{r}} + \ldots} & {2{nd}\mspace{14mu} {shelf}}\end{matrix} \right.$

One Possible Embodiment

The equations for one possible embodiment of the invention involve a onekilogram mass, and a restriction of the angular velocity to less than800 revolutions per minute (rpm). The mass will be comprised of smallspherical pieces of lead or similarly dense material. The center of thefirst bowl of the device will contain a vertically oriented axle with adiameter of 0.08 meter for attachment to the wind turbine. Radius (r)and height (h) are measured in meters. The function describing the bowlsand shelves of the device is:

${h(r)} = \left\lbrack \begin{matrix}{{{.04} \leq r \leq {.072}},} & {15927 \cdot \left( {r^{4} - {{.08}r^{3}} + {{.0016}r^{2}}} \right)} & {{first}\mspace{14mu} {bowl}} \\{{{.072} < r \leq {.11}},} & {{{.0001} \cdot \sqrt{r}} + {h({.072})}} & {{first}\mspace{14mu} {shelf}} \\{{{.11} < r \leq {.14}},} & {{2548.42 \cdot \left( {r^{4} - {{.11527}r^{3}} + {{.003322}r^{2}}} \right)} + {h({.11})}} & {2{nd}\mspace{14mu} {bowl}} \\{{{.14} < r \leq {.182}},} & {{{.001} \cdot \sqrt{r}} + {h({.14})}} & {2{nd}\mspace{14mu} {shelf}} \\{{{.192} < r \leq {.222}},} & {{637.105 \cdot \left( {r^{4} - {{.11058}\mspace{11mu} r^{3}} + {{.003057}\mspace{11mu} r^{2}}} \right)} + {h({.182})}} & {3{rd}\mspace{14mu} {bowl}}\end{matrix} \right.$

The drawing of this embodiment is set forth later in this application inFIG. 2. A sampling of dynamic values associated with this embodiment isset forth in the following table.

Angular Moment of Kinetic velocity Inertia Energy Radius (m) Height (m)(rpm) (kg · m²) (joules) First Bowl .04 0 0 .00160 0 .048 .002349 191.00230 0.461 .056 .012787 382 .00314 2.51 .064 .037578 573 .00410 7.37.072 .084550 764 .00518 16.6 First Shelf .080 .084579 688 .00640 16.6.088 .084580 625 .00774 16.6 .096 .084581 573 .00922 16.6 .104 .084582529 .0108 16.6 2nd Bowl .112 .094475 519 .0125 18.5 .120 .142721 596.0144 28.0 .128 .206719 672 .0164 40.6 .136 .289449 749 .0185 56.8 2ndShelf .144 .33922 765 .0207 66.5 .152 .33923 725 .0231 66.5 .160 .33924689 .0256 66.5 .168 .33925 656 .0282 66.5 .176 .33926 626 .0310 66.5 3rdBowl .184 .3573 615 .0339 70.1 .192 .4389 653 .0369 86.1 .200 .534 691.0400 105 .208 .643 729 .0433 126 .216 .768 768 .0467 151 .222 .873 796.0493 171

One advantage of the invention becomes clear from these figures. Ofcourse, with a fixed momentum flywheel, as angular velocity increases,so does kinetic energy. However, with the present invention, at eachshelf angular velocity decreases while kinetic energy remainsapproximately constant because the moment of inertia simultaneouslyincreases. This allows the device to spin within a particular range ofangular velocity while kinetic energy continues to increase in astep-wise manner, similar to the effect of an automatic transmission ina motor vehicle. The resulting benefits include: 1) increased energystorage, 2) regulation of angular velocity, and 3) a method ofpreventing excessive angular velocity and the consequential “stall” oflift type vertical axis wind turbines.

1-10. (canceled)
 11. A flywheel device (8) a moment of inertia of whichis varied by automatically redistributing a fluid, said fluid being aliquid, set of small spherical, or non-spherical solid particles, withinthe flywheel according to changes in angular velocity of the flywheel bymeans of a plurality of alternating and concentric bowls (3, 5, 7) andshelves (4, 6), an empty interior space (13) defined by the bowls andshelves divided by radially oriented vertical walls (14), the bowlsshaped concave up to allow the said fluid to climb the bowl due toincreasing said angular velocity and to fall down the bowl due todecreasing said angular velocity, and the shelves shaped concave down toallow the fluid to flow to an outer edge of the shelf thereby decreasingsaid angular velocity and to flow to the inner parts of the shelfthereby increasing said angular velocity.
 12. The flywheel deviceaccording to claim 11 in which the concavity of the bowls is greaterthan that determined by a function${{h(r)} = {{r\mspace{11mu} {\tan^{- 1}({kr})}} - {\frac{1}{2}{\ln \left( {1 + {k^{2}r^{2}}} \right)}} + C}},$where h is a vertical distance from a surface of the bowl to an innerrim of the bowl (10) or to a hub (11) if a first bowl, r is the radialdistance, k is a constant equal to 1, and C is a constant equal to zero.13. The flywheel device according to claim 12 in which k is a constantof any positive value, and C is a constant of any value.